一类随机规划逼近最优值和最优解集的收敛性

本文提出强上图收敛的概念,讨论了逼近随机规划的目标函数序列的强上图收敛性,研究了逼近随机规划最优值和最优解集的收敛性条件,得到了一类随机规划逼近最优值和最优解集的收敛性.     

关于任意非负随机序列随机和的一类随机偏差定理

本文引入任意随机变量序列随机极限对数似然比概念,作为任意相依随机序列联合分布与其边缘乘积分布“不相似”性的一种度量,利用构造新的密度函数方法来建立几乎处处收敛的上鞅,在适当的条件下,给出了任意受控随机序列的一类随机偏差定理.     

极大极小随机规划逼近问题最优解集和最优值的稳定性

研究了特殊的二层极大极小随机规划逼近收敛问题. 首先将下层初始随机规划最优解集拓展到非单点集情形, 且可行集正则的条件下, 讨论了下层随机规划逼近问题最优解集关于上层决策变量参数的上半收敛性和最优值函数的连续性. 然后把下层随机规划的epsilon-最优解向量函数反馈到上层随机规划的目标函数中, 得到了上层随机规划逼近问题的最优解集关于最小信息概率度量收敛的上半收敛性和最优值的连续性.     

二层随机规划逼近解的收敛性

对二层随机规划的逼近解的收敛性作了探讨,证明了当随机向量序列{ζ(k)(w)}依分布收敛于ζ(w)时,相应于ζ(k)(w)的二层随机规划问题的任何最优解序列将收敛到原问题的最优解.     

二层随机规划逼近问题最优解集的上半收敛性

在下层初始随机规划问题可行解集上引入了正则的概念,并在下层初始随机规划最优解唯一的条件下,利用上图收敛理论,给出了下层随机规划逼近问题的任意一个最优解向量函数都连续收敛到下层初始随机规划问题的唯一最优解向量函数.然后将下层随机规划的最优解向量函数反馈到上层随机规划的目标函数和约束条件中,得到了上层随机规划逼近问题的最优解集关于最小信息概率度量收敛的上半收敛性.     

随机Bernstein多项式的依分布收敛问题

利用随机的Bernstein多项式研究随机逼近问题具有一定的意义.借助弱收敛的概念,从分布函数的角度,讨论了随机Bernstein多项式依分布收敛问题.同时,与依概率收敛结果相比较,以此说明Bernstein多项式序列依分布收敛适用的范围更广.     

二层随机规划逼近最优解集的上半收敛性

下层随机规划以上层决策变量作为参数,而上层随机规划是以下层随机规划的唯一最优解作为响应的一类二层随机规划问题,首先在下层随机规划的原问题有唯一最优解的假设下,讨论了下层随机规划的任意一个逼近最优解序列都收敛于原问题的唯一最优解,然后将下层随机规划的唯一最优解反馈到上层,得到了上层随机规划逼近最优解集序列的上半收敛性.     

一个求概率约束规划的随机拟次梯度算法

概率约束规划是经常费到的一类规划，但其约束函数含有概率，在一般场合下，很难求出，随机拟次梯度法无须计算约束值与导数值，只要构造出约束函数目标函数的随机拟次梯度即可，本文给出了一个求解概率约束规划的随机拟次梯度算法，并证明了有关的定量及性质。     

概率约束规划的稳定性分析

本文对概率约束规划问题的稳定性进行了探讨,得出了当随机向量序列{ξ^(k)(ω）}分布收敛于ξ（ω）时,相应于ξ^(k)(ω)的概率约束规划问题的最优值收敛于原问题的最优值,这个结果为设计逼近算法和改进逼近解提供了一个理论基础。     

随机环境中马氏链转移函数的收敛性

本文研究了随机环境中马氏链转移函数的收敛性．利用Doeblin条件，得到了随机环境中一般状态马氏链转移函数弱收敛及强收敛的一些条件，以及收敛速度的一个估计．     

概率约束规划逼近最优解集的上半收敛性

本文讨论了概率约束规划目标函数的连续收敛性,并利用概率测度弱收敛的特征给出了概率约束规划可行集的收敛性条件,得到了概率约束规划逼近最优解集的上半收敛性.     

关于任意相依随机序列的完全收敛性(英文)

引入极限对数似然比的概念作为任意随机序列的联合分布与其边缘分布的差异的随机性度量,用概率密度比构造几乎处处收敛的上鞅,在适当的条件下,给出任意随机序列完全收敛的若干定理.     

Approximations for chance-constrained programming problems

This paper proposes an approximation approach to the solution of chance-constrained stochastic programming problems. The results rely in a fundamental way on the theory of convergence of sequences of measurable multifunctions. Particular results are presented for stochastic linear programming problems.     

Convergence Properties of Two-Stage Stochastic Programming

This paper considers a procedure of two-stage stochastic programming in which the performance function to be optimized is replaced by its empirical mean. This procedure converts a stochastic optimization problem into a deterministic one for which many methods are available. Another strength of the method is that there is essentially no requirement on the distribution of the random variables involved. Exponential convergence for the probability of deviation of the empirical optimum from the true optimum is established using large deviation techniques. Explicit bounds on the convergence rates are obtained for the case of quadratic performance functions. Finally, numerical results are presented for the famous news vendor problem, which lends experimental evidence supporting exponential convergence.     

概率约束随机规划的一种近似方法及其它的有效解模式

根据最小风险的投资最优问题，我们给出了一个统一的概率约束随机规划模型。随后我们提出了求解这类概率约束随机规划的一种近似算法，并在一定的条件下证明了算法的收敛性。此外，提出了这种具有概率约束多目标随机规划问题的一种有效解模型。     

Normalized convergence in stochastic optimization

A new concept of (normalized) convergence of random variables is introduced. This convergence is preserved under Lipschitz transformations, follows from convergence in mean and itself implies convergence in probability. If a sequence of random variables satisfies a limit theorem then it is a normalized convergent sequence. The introduced concept is applied to the convergence rate study of a statistical approach in stochastic optimization.     

Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems

Several exponential bounds are derived by means of the theory of large deviations for the convergence of approximate solutions of stochastic optimization problems. The basic results show that the solutions obtained by replacing the original distribution by an empirical distribution provides an effective tool for solving stochastic programming problems.Supported in part by a grant from the US-Israel Science Foundation.     

An SQP algorithm for extended linear-quadratic problems in stochastic programming

Extended Linear-Quadratic Programming (ELQP) problems were introduced by Rockafellar and Wets for various models in stochastic programming and multistage optimization. Several numerical methods with linear convergence rates have been developed for solving fully quadratic ELQP problems, where the primal and dual coefficient matrices are positive definite. We present a two-stage sequential quadratic programming (SQP) method for solving ELQP problems arising in stochastic programming. The first stage algorithm realizes global convergence and the second stage algorithm realizes superlinear local convergence under a condition calledB-regularity.B-regularity is milder than the fully quadratic condition; the primal coefficient matrix need not be positive definite. Numerical tests are given to demonstrate the efficiency of the algorithm. Solution properties of the ELQP problem underB-regularity are also discussed.Supported by the Australian Research Council.     

Epigraphical nesting: A unifying theory for the convergence of algorithms

In an earlier paper, the authors introduced epigraphical nesting of objective functions as a means to characterize the convergence of global optimization algorithms. Epigraphical nesting of objective functions may be looked upon as a relaxation of epigraphical convergence of objective functions, whereby one ensures that epigraphs of the approximations contain asymptotically the epigraph of the objective function of the original optimization problem. In this paper, we show that, for algorithms which seek only a stationary point, convergence can be assured by objective function approximations whose directional derivatives attain an epigraphical nesting property. We demonstrate that epigraphical nesting provides a unifying thread that ties together a number of different algorithms, including those for the solution of variational inequalities and smooth as well as nonsmooth optimization. We show that the Newton method and its variants for unconstrained optimization, successive quadratic programming methods for constrained optimization, and proximal point algorithms (deterministic and stochastic) all construct approximations in which the directional derivatives attain an epigraphical nesting property, even though the approximations themselves fail to attain such a property.This work was supported in part by Grants NSF-DDM-89-10046 and NSF-DDM-91-14352 from the National Science Foundation.     

任意随机序列的变换及其a.s.收敛性

本文利用截尾方法构造几乎处处收敛的鞅结合无穷乘积定理，研究随机变量序列变换的局部收敛性及强大数定理，作为推论得到了关于赌博系统的若干强极限定理。